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## G = S3×A5order 360 = 23·32·5

### Direct product of S3 and A5

Aliases: S3×A5, C3⋊(C2×A5), (C3×A5)⋊1C2, SmallGroup(360,121)

Series: ChiefDerived Lower central Upper central

 Chief series C1 — C3 — S3 — S3×A5
 Derived series A5 — C3×A5 — S3×A5
 Lower central C3×A5 — S3×A5
 Upper central C1

3C2
15C2
45C2
10C3
20C3
6C5
5C22
45C22
45C22
10S3
15C6
15S3
30S3
30C6
60S3
10C32
6D5
18D5
18C10
6C15
15C23
5A4
10A4
15D6
15D6
30D6
10C3×S3
10C3×S3
10C3⋊S3
18D10
6D15
15C2×A4
10S32

Character table of S3×A5

 class 1 2A 2B 2C 3A 3B 3C 5A 5B 6A 6B 10A 10B 15A 15B size 1 3 15 45 2 20 40 12 12 30 60 36 36 24 24 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 1 1 1 1 1 -1 -1 -1 1 1 linear of order 2 ρ3 2 0 2 0 -1 2 -1 2 2 -1 0 0 0 -1 -1 orthogonal lifted from S3 ρ4 3 -3 -1 1 3 0 0 1-√5/2 1+√5/2 -1 0 -1-√5/2 -1+√5/2 1+√5/2 1-√5/2 orthogonal lifted from C2×A5 ρ5 3 3 -1 -1 3 0 0 1+√5/2 1-√5/2 -1 0 1-√5/2 1+√5/2 1-√5/2 1+√5/2 orthogonal lifted from A5 ρ6 3 3 -1 -1 3 0 0 1-√5/2 1+√5/2 -1 0 1+√5/2 1-√5/2 1+√5/2 1-√5/2 orthogonal lifted from A5 ρ7 3 -3 -1 1 3 0 0 1+√5/2 1-√5/2 -1 0 -1+√5/2 -1-√5/2 1-√5/2 1+√5/2 orthogonal lifted from C2×A5 ρ8 4 -4 0 0 4 1 1 -1 -1 0 -1 1 1 -1 -1 orthogonal lifted from C2×A5 ρ9 4 4 0 0 4 1 1 -1 -1 0 1 -1 -1 -1 -1 orthogonal lifted from A5 ρ10 5 -5 1 -1 5 -1 -1 0 0 1 1 0 0 0 0 orthogonal lifted from C2×A5 ρ11 5 5 1 1 5 -1 -1 0 0 1 -1 0 0 0 0 orthogonal lifted from A5 ρ12 6 0 -2 0 -3 0 0 1-√5 1+√5 1 0 0 0 -1-√5/2 -1+√5/2 orthogonal faithful ρ13 6 0 -2 0 -3 0 0 1+√5 1-√5 1 0 0 0 -1+√5/2 -1-√5/2 orthogonal faithful ρ14 8 0 0 0 -4 2 -1 -2 -2 0 0 0 0 1 1 orthogonal faithful ρ15 10 0 2 0 -5 -2 1 0 0 -1 0 0 0 0 0 orthogonal faithful

Permutation representations of S3×A5
On 15 points - transitive group 15T23
Generators in S15
(1 2 3 4 5)(6 7 8 9 10 11 12 13 14 15)
(1 15 10)(2 13 14)(3 7 12)(4 9 6)(5 11 8)

G:=sub<Sym(15)| (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,15,10)(2,13,14)(3,7,12)(4,9,6)(5,11,8)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15), (1,15,10)(2,13,14)(3,7,12)(4,9,6)(5,11,8) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10,11,12,13,14,15)], [(1,15,10),(2,13,14),(3,7,12),(4,9,6),(5,11,8)])

G:=TransitiveGroup(15,23);

On 18 points - transitive group 18T145
Generators in S18
(2 3)(4 5 6 7 8)(9 10 11 12 13 14 15 16 17 18)
(1 11 10)(2 16 7)(3 8 15)(4 13 14)(5 9 12)(6 17 18)

G:=sub<Sym(18)| (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,11,10)(2,16,7)(3,8,15)(4,13,14)(5,9,12)(6,17,18)>;

G:=Group( (2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18), (1,11,10)(2,16,7)(3,8,15)(4,13,14)(5,9,12)(6,17,18) );

G=PermutationGroup([(2,3),(4,5,6,7,8),(9,10,11,12,13,14,15,16,17,18)], [(1,11,10),(2,16,7),(3,8,15),(4,13,14),(5,9,12),(6,17,18)])

G:=TransitiveGroup(18,145);

On 30 points - transitive group 30T85
Generators in S30
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 27 17)(2 14 26)(3 29 13)(4 16 28)(5 25 15)(6 20 24)(7 21 19)(8 12 30)(9 23 11)(10 18 22)

G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,27,17)(2,14,26)(3,29,13)(4,16,28)(5,25,15)(6,20,24)(7,21,19)(8,12,30)(9,23,11)(10,18,22)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,27,17)(2,14,26)(3,29,13)(4,16,28)(5,25,15)(6,20,24)(7,21,19)(8,12,30)(9,23,11)(10,18,22) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,27,17),(2,14,26),(3,29,13),(4,16,28),(5,25,15),(6,20,24),(7,21,19),(8,12,30),(9,23,11),(10,18,22)])

G:=TransitiveGroup(30,85);

On 30 points - transitive group 30T94
Generators in S30
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 26 12)(2 17 25)(3 28 16)(4 11 27)(5 13 20)(6 24 18)(7 22 23)(8 30 21)(9 15 29)(10 19 14)

G:=sub<Sym(30)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,26,12)(2,17,25)(3,28,16)(4,11,27)(5,13,20)(6,24,18)(7,22,23)(8,30,21)(9,15,29)(10,19,14)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,26,12)(2,17,25)(3,28,16)(4,11,27)(5,13,20)(6,24,18)(7,22,23)(8,30,21)(9,15,29)(10,19,14) );

G=PermutationGroup([(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,26,12),(2,17,25),(3,28,16),(4,11,27),(5,13,20),(6,24,18),(7,22,23),(8,30,21),(9,15,29),(10,19,14)])

G:=TransitiveGroup(30,94);

On 30 points - transitive group 30T102
Generators in S30
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)
(1 3 29)(4 13 28)(5 20 12)(6 17 19)(7 21 16)(8 10 30)(14 25 27)(15 22 24)

G:=sub<Sym(30)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30), (1,3,29)(4,13,28)(5,20,12)(6,17,19)(7,21,16)(8,10,30)(14,25,27)(15,22,24) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30)], [(1,3,29),(4,13,28),(5,20,12),(6,17,19),(7,21,16),(8,10,30),(14,25,27),(15,22,24)])

G:=TransitiveGroup(30,102);

Polynomial with Galois group S3×A5 over ℚ
actionf(x)Disc(f)
15T23x15+5x14+8x13+7x12+8x11-3x9+7x8-2x7+7x6-2x5-3x4+5x3-2x+1-237·416·4632

Matrix representation of S3×A5 in GL5(𝔽31)

 0 1 0 0 0 1 0 0 0 0 0 0 3 29 21 0 0 2 10 3 0 0 1 0 0
,
 0 30 0 0 0 1 30 0 0 0 0 0 28 21 29 0 0 29 3 10 0 0 30 0 0

G:=sub<GL(5,GF(31))| [0,1,0,0,0,1,0,0,0,0,0,0,3,2,1,0,0,29,10,0,0,0,21,3,0],[0,1,0,0,0,30,30,0,0,0,0,0,28,29,30,0,0,21,3,0,0,0,29,10,0] >;

S3×A5 in GAP, Magma, Sage, TeX

S_3\times A_5
% in TeX

G:=Group("S3xA5");
// GroupNames label

G:=SmallGroup(360,121);
// by ID

G=gap.SmallGroup(360,121);
# by ID

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